Problem: The following system of equations is represented by the matrix equation $\text{A}\left[\begin{array} {ccc} x \\ y \\ z \end{array} \right]=\vec{b}$. $\begin{aligned}-8x+6y+2z&=22 \\3x+5y+9z&=-4 \\12x+16y+8z&=10\end{aligned}$ ${A}=$ $\vec{b}=$ Represent each row and column in the order in which the variables and equations appear.
Explanation: The Strategy A system of equations can be represented by a matrix equation $\text{A}\vec{x}=\vec{b}$, where $\text{A}$ is the coefficient matrix, $\vec{x}$ is the variables vector, and $\vec{b}$ is the constants vector. Each row of the matrix equation represents an equation in the system. [I need an explanation, please!] Representing the system of equations as a matrix equation We are given the system of equations: $\begin{aligned}-8x+6y+2z&=22 \\3x+5y+9z&=-4 \\12x+16y+8z&=10\end{aligned}$ First, let's rewrite this system to show the coefficients of each variable. $\begin{aligned}{-8}x+{6}y+{2}z&=22 \\{3}x+{5}y+{9}z&=-4 \\{12}x+{16}y+{8}z&=10\end{aligned}$ Now, the coefficient matrix can be written as follows. $\left[\begin{array} {ccc} {-8} & {6} & {2} \\ {3} & {5} & {9} \\ {12} & {16} & {8} \end{array} \right]$ We can multiply this matrix by a column vector of variables and set it equal to a column vector with the values on the right side of the equations, as follows. $\left[\begin{array} {ccc} {-8} & {6} & {2} \\ {3} & {5} & {9} \\ {12} & {16} & {8} \end{array} \right]\left[\begin{array} {ccc} x \\ y \\ z \end{array} \right] =\left[\begin{array} {ccc} 22 \\ -4 \\ 10 \end{array} \right]$ This is our matrix equation. Summary $\text{A}$ and $\vec{b}$ are shown below. $\text{A}=\left[\begin{array} {ccc} -8 & 6 & 2 \\ 3 & 5 & 9 \\ 12 & 16 & 8 \end{array} \right]~~~~~~~~~~~~ \vec{b}=\left[\begin{array} {ccc} 22 \\ -4 \\ 10\end{array} \right]$